"You have added a new aspect of informal summit in the bilateral relationship which I think is a great occasion and creates trust," Modi said.
"The Monday talks were very intense," Russia's official Tass news agency quoted Foreign Minister Sergei Lavrov as saying.
"I am confident that the ongoing informal contacts between the leaders of Russia and India will be useful and will help define the further guidelines for our development and strategic partnership," he said.
"There was a discussion of various areas of our specific cooperation at the regional and global level," Lavrov said.
"We discussed the whole spectrum of our particularly privileged strategic partnership, paid special attention to the economy, noted the steady growth of trade turnover," the minister said.
"We are working together on International North-South Transport Corridor (INSTC) and BRICS," Modi said. He also congratulated Putin for becoming the President for the fourth time with overwhelming majority.
Putin said his visit would give a fresh impetus to bilateral ties. "Our defense ministries maintain very close contacts and cooperation. It speaks about a very high strategic level of our partnership," Putin said.
He also hailed joint activities of the two countries in the area of foreign politics, in particular within the United Nations, BRICS and the SCO.
The CAATSA is a United States federal law that imposed sanctions on Iran, North Korea, and Russia. It includes sanctions against countries that engage in significant transactions with Russia's defence and intelligence sectors.
India is not going to allow its defence engagement with Russia to be dictated by any other country, the sources said, adding New Delhi has been lobbying with the Trump administration on the issue.
India could face US sanctions for purchasing high value military defence items, in particular state-of-the-art S-400 Triumf missile defence system, from Russia under the act.
In his opening remarks, Modi also thanked Russia for playing a major role in helping India get a permanent membership in the Shanghai Cooperation Organisation. India and Pakistan were admitted last year into the organisation.
"Starting from the year 2000 when you took office, our relation has been historic...," Modi said.
He said Russia and India maintain a high strategic level of partnership, with close cooperation between the two countries' defense ministries.
"There is no need to explain Russian-Indian relations, since we all know that they have deep roots. However, we have been able to create additional momentum recently. Last year, our trade saw a significant increase, adding another 17 percent since the beginning of this year," he said.
The Russian Foreign Minister also said that Moscow and New Delhi will strongly oppose attempts to politicise the work of the Organization for the Prohibition of Chemical Weapons (OPCW).
Russia and India had confirmed they "will strongly oppose attempts to politicise the work of the Organization for the Prohibition of Chemical Weapons," Lavrov said.
He pointed out that "we have seen" such attempts "recently." His comments came following the OPCW last month said it had confirmed Britain's findings on the military-grade nerve agent used in the attack on ex-Russian spy Sergei Skripal and his daughter Yulia in the UK, which London said came from Russia.
Later, President Putin hosted a lunch in the honour of Prime Minister Modi and the two leaders also took a boat ride from Bocharev Creek to Olympic Park in Russia's Sochi.
Modi, Putin take boat ride in Sochi
Prime Minister Narendra Modi and Russian President Vladimir Putin took a boat ride from Bocharev Creek to Olympic Park in Russia's Sochi after their informal summit so they can have more time to discuss bilateral ties.
The two leaders are in this Black Sea coastal city for their first informal summit.
The talks between the two leaders in Putin's Bocharov Ruchei residence lasted for more than three hours. After the official part, the two leaders continued informal talks. Putin invited Modi for a boat ride, Russia's official news agency Tass reported.
"Boat ride on the Black Sea! PM @narendramodi and Russian President #Putin had plenty to discuss as they sail together from Bocharev Creek to Olympic Park in #Sochi," External Affairs Ministry spokesperson Raveesh Kumar tweeted.
During the sea trip, they spoke in English without the support of an interpreter. The leaders continued negotiations at lunch, going out on the deck from time to time to breathe in the fresh sea air and marvel at the coastal line of Sochi that had been refurbished before the Winter Olympic Games it hosted in 2014.
Earlier in the day, Modi during his talks with the Russian president underlined the strong bond between the two "strategic partners".
Prime Minister Narendra Modi also congratulated President Vladimir Putin and Russia for playing host to the FIFA World Cup 2018 beginning next month and hoped for the successful organisation of the sporting extravaganza.
The 2018 men's FIFA World Cup will be held in Russia from June 14 to July 15. A total of 64 matches will be played by 32 teams in 12 venues located in 11 cities.
The final will be held at Luzhniki Stadium in Moscow.
Prime Minister Modi, who is here for the first informal summit with Putin, said that the world is eagerly waiting for the sporting event.
"The entire world is eagerly waiting for the FIFA World Cup 2018 and you are going to organise the event. You will be at the centre of the world's attention and I congratulate you for that," Modi said.
The international soccer competition takes place every four years. For the first time in 32 years, the US will not compete in the World Cup as its men's national
team has failed to qualify.